On the Number of Prime Factors of Consecutive Integers

Abstract: We prove that there are infinitely many $n$ such that $ω(n+k) \ll \log k$ for all integers $k \ge 2$. This improves on a result of Tao-Teräväinen (2025), who has $O(k)$ in place of $O(\log k)$. As corollaries, we make progress on a number of questions posed by Erdős. The proof is based on a quantitative refinement of the Tao-Teräväinen probabilistic argument, combining a more efficient sieve procedure with stronger exponential concentration-of-measure estimates. Moreover, we formulate a conjecture on integers with many prime factors based on Cramér-type random models. Assuming this conjecture, the main bound is essentially sharp.

• The paper proves that there exist infinitely many n for which every consecutive integer n+k has its prime factors count ω(n+k) bounded by C log k.
• It employs a refined randomized sieve and concentration of measure techniques to derive exponential tail bounds for the distribution of prime factors.
• These results resolve several Erdős conjectures by establishing nearly optimal logarithmic bounds, advancing our understanding of additive prime factor structures.

Prime Factors in Consecutive Integers: Quantitative Refinements and Implications

Introduction and Main Results

This work investigates the upper bound on the number of prime factors $\omega(n+k)$ (distinct prime factors) and $\Omega(n+k)$ (counting multiplicities) that can simultaneously occur for long sequences of consecutive integers. The central result establishes that there exist infinitely many $n$ such that $\omega(n+k) \leq C \log k$ holds for all $k \geq 2$, improving on the recent result of Tao–Teräväinen, which only achieved an $O(k)$ bound (Tao et al., 1 Dec 2025). This sharp logarithmic refinement approaches the threshold suggested by probabilistic Cramér-type models.

Explicitly, the author proves:

For some absolute constant $C > 0$, there are infinitely many positive integers $n$ such that $\omega(n+k) \leq C\log k$ for all $k \geq 2$.

The same holds for the backward shifts ($\Omega(n+k)$0), and similar bounds apply to the total number of prime factors $\Omega(n+k)$1.

Additionally, corollaries improve several long-standing conjectures of Erdős on the distribution of small-prime-factor integers across consecutive sequences, including weakened forms on the divisor function $\Omega(n+k)$2 and on the maximal number of prime factors of $\Omega(n+k)$3. The techniques also yield conditional (via Cramér models) evidence that the established bounds are nearly optimal up to constant factors in $\Omega(n+k)$4.

Technical Framework and Innovations

The proof framework synthesizes quantitative probabilistic arguments with delicate sieve-theoretic constructions modeled on the Maynard–Tao type, but with crucial quantitative optimizations:

• Randomized Sieve Construction: The argument uses a highly structured weighted sieve procedure on $\Omega(n+k)$5, imposing divisibility and non-divisibility conditions (via high-multiplicity congruence restrictions and modern Selberg–Goldston-Pintz–Yıldırım weights) on each shift $\Omega(n+k)$6 for $\Omega(n+k)$7. The sieving levels $\Omega(n+k)$8 are optimally decayed polynomially in $\Omega(n+k)$9 to balance two crucial constraints: keeping product of sieve levels below $n$0 for some $n$1, and ensuring typical behavior of $n$2 closely tracks the mean for "smooth" integers.
• Concentration of Measure: The heart of the improvement is a precise analysis of the large deviations for $n$3 under the weighted model. The author establishes strong exponential concentration—quantitative analogs of the central limit theorem—so that the probability $n$4 drops off as $n$5 (for some positive $n$6), allowing the union bound over all $n$7. Previous approaches yielded only exponential bounds in $n$8, leading to linear dependencies.
• High-Moment and Tail Estimates: The proof executes careful moment calculations and incorporates Stirling number combinatorics to bound higher moments (up to $n$9), enabling tail bounds essential for converging the global union bounds.
• Sharp Control of Exceptional Cases: For large $\omega(n+k) \leq C \log k$0 ($\omega(n+k) \leq C \log k$1), the growth rate of $\omega(n+k) \leq C \log k$2 is trivially controlled by standard $\omega(n+k) \leq C \log k$3 estimates, so the main effort concentrates on the "near shift" regime.

Contradictions to Erdős Conjectures via Probabilistic Models

Employing Cramér-type random models and the density estimates for integers with prescribed numbers of prime factors [tenenbaum2015introduction], the paper formulates a conjecture: for $\omega(n+k) \leq C \log k$4 large enough, there always exists some $\omega(n+k) \leq C \log k$5 such that $\omega(n+k) \leq C \log k$6, making the proven $\omega(n+k) \leq C \log k$7 essentially optimal.

Conditionally, this implies that Erdős' strongest conjecture (Problem #679), positing $\omega(n+k) \leq C \log k$8 for all $\omega(n+k) \leq C \log k$9, cannot hold—random models predict occasional much larger deviations.

Implications and Theoretical Significance

Practical and Theoretical Impact

• Sharpness of Sieve Refinements: This result pushes the limits of multidimensional sieve methods for controlling additive shifts, demonstrating that the "probabilistic parity barrier" in such settings is essentially at the $k \geq 2$0 regime.
• Resolution of an Erdős Problem: It resolves previously open conjectures on the simultaneous smallness of $k \geq 2$1 across all $k \geq 2$2, providing a strong counterpoint to the belief that "arbitrarily long runs" of $k \geq 2$3-almost-primes could exhibit only sublogarithmic growth in their number of factors.
• Framework for General Patterns: The technical machinery is applicable to a range of shifted convolution problems involving divisor or prime factor counts and immediately generalizes to related questions (e.g., concerning $k \geq 2$4, $k \geq 2$5).

Directions for Future Research

• Optimality and Random Model Deviations: Determining whether the logarithmic constant is best possible unconditionally remains open. Further study might connect finer sieve balance to analytic properties of Dirichlet polynomials associated with the problem, in line with Maynard’s optimal polynomial sieve (Ford et al., 2024).
• Analogous Results for Other Multiplicative Functions: The approach could be adapted to the shifted distribution of other arithmetic functions, such as sum-of-divisors or highly composite numbers, particularly in contexts where correlations over shifts are delicate.
• Distribution in Short Intervals and Local Gaps: The author’s probabilistic bounding technique suggests further application in local gap and pattern detection for almost-prime sequences, for which unconditional results are generally sparse.

Conclusion

This work establishes the existence of long runs of consecutive integers, each with at most $k \geq 2$6 prime factors, for all $k \geq 2$7 and infinitely many base points $k \geq 2$8. The main technical innovation is a refined probabilistic analysis of large sieve-based random models, yielding exponential moment estimates sufficient to trump previous methods and essentially match the probabilistic lower bound. The results advance our understanding of the fine-scale additive structure of prime divisors and clarify the inherent limits of sieve-based approaches to consecutive almost-primes, settling several conjectures and drawing new dichotomies with classical random models. The techniques introduced have immediate applicability to broader quantitative questions in analytic and probabilistic number theory.

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Reference:

Cheuk Fung (Joshua) Lau, "On the Number of Prime Factors of Consecutive Integers" (2604.15042).